Each harmonic was described by four Fourier coefficients, two each for the x -and y -axes, generating a total of 4ncoefficients labeled a_n, b_n, c_n and d_n, where n is the number of harmonics. They combine to describe the repeated elements in a sinusoidal waveform, (aj , bj ) and (cj , dj ) were the four Fourier coefficients defining each harmonic (j th order); k corresponds to the maximum number of harmonics used for the Fourier decompo­sition and T was equal to the perimeter of the outline. An incremental number of harmonics in Fourier series converges on detailed information on the morphology of analyzed form, and numerically describes distortions and complexity of forms (from the original circle) described by the Fourier series when no harmonics are present. In this manner, Elliptical Fourier Analysis (EFA) is able to resolve and describe complex 2-Dimensional bounded outlines and also spatial orientation relation to reference planes (Ferrario et al., 1996), allometric changes and shape (Procrustes coordinates) can be assessed when plotted against CS. The size normalization procedure consisted of a recalculation of the outlines using the same value of the enclosed area for all specimens.

Measurement error was estimated from a Procrustes ANOVA by considering individual as the main source of variation and residuals rep­resenting variation in digitized replicates as a second source of error (Cocilovo et al. 2012). To remove differences due to specimen orientation and position during data collection, and to separate size and shape components; landmark configurations are first scaled to the same size, centered at their origin and rotated to minimize the distances among the corresponding landmarks (Generalized Procrustes Analysis or GPA) (Rohlf, 2010). GPA superimposes specimen landmark configurations by translating them to a common origin, scaling them to unit centroid size (the square root of the sum of squared distances of all landmarks to the centroid of the object), and rotating them according to a best-fit criterion

Figure 2a and b Showing dorsal, ventral and direct caudal views of the small African Pangolin skull (P. tricuspis ) and landmarks assessed 1. Posthion in the midline of the asal bone, 2. Suture of the nasal bone in the midline rostral limit of the frontal bone, 3 Parietal suture in the midline at the rostral limit of the parietal bone, 4 the Inter-parietal bone suture at the caudal limit of the parietal in the midline, 5 Dorsal limit of the foramen magnum in the midline of the occipital bone, 6 and 7 Lateral boundaries of the nasal bone sutures in the maxilla, 8 and 9 Lateral limits of the parietal bone sutures at the temporal bones, 10 and 11 Lateral limits of the Inter-parietal bone sutures with the occipital bone. The ventral view landmarks includes; 1 the Posthion in the ventral view, 2 the ventral midline boundary of the sphenoid, 3 Ventral rim of the foramen magnum in the ventral midline, 4 and 5 Ventral tips of the zygoma on both sides, 6 and 7 Ventro-lateral limits of the temporal bone on both sides, 8 and 9 The lateral boundaries of the occipital condyles in ventral view (b) the caudal view of foramen magnum outline

Size normalization of descriptor coefficients of foramen outlines was achieved by the ellipse of the first harmonic described for closed shapes (Kuhl and Giardina, 1982) to be invariant of size employing CHc 2-NEF SHAPE Version 1.3. Major axis length/2, Minor axis length/2, orientation of the major axis and phase angle (θ = 1/2 arct.2 (a₁b₁ + c₁d₁)/a₁²+ c₁² – b₁² – d₁²) corresponding to the position elliptical first point are the four descriptors of each harmonic. A stepwise Elliptical Fourier reconstruction of foramen magnum outlines was done employing incremental number of harmonics. Size and peculiar anisotropic characteristics were determined for each harmonic (Major axis length/2 x Minor axis length/2 and Major axis/Minor axis respectively) (Buck, 1962).

Descriptor coefficients were analyzed with ‘PrinComp’ based on Variance-Co­variance matrix of normalized coefficients. It was also noted that such coefficients with small variance and covariation values do not significantly explain mor­phological variations and are now calculated and used in principal components derivation bearing all edge/contour shapes information in the first 14 harmonics as explained by Rholf and Archie, (1984). Summary statistics showing mean ± SD employing PAST (Hammer et al., 2013) for foramen magnum shape outline descriptors was performed. An allometric (shape vs size) based discriminant function analysis was evaluated with incremental Fourier descriptor harmonics followed by use of inverse Fourier transform to obtain graphical representation at 0.05 Bonferroni post-test level of significance.